\(\int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) [804]

Optimal result

Integrand size = 22, antiderivative size = 445 \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 a x^5}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (13 b c-7 a d) x^4}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+14 a b c d-7 a^2 d^2\right ) x^3 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {2 c (b c+a d) \left (7 b^2 c^2-22 a b c d+7 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right )-2 b d \left (35 b^4 c^4-76 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-76 a^3 b c d^3+35 a^4 d^4\right ) x\right )}{12 b^4 d^4 (b c-a d)^4}+\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{9/2} d^{9/2}} \]

[Out]

2/3*a*x^5/b/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)+5/4*(7*a^2*d^2+10*a*b*c*d+7*b^2*c^2)*arctanh(d^(1/2)*(b*x+a
)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(9/2)/d^(9/2)+2/3*a*(-7*a*d+13*b*c)*x^4/b^2/(-a*d+b*c)^2/(d*x+c)^(3/2)/(b*x+a
)^(1/2)-2/3*c*(-7*a^2*d^2+14*a*b*c*d+b^2*c^2)*x^3*(b*x+a)^(1/2)/b^2/d/(-a*d+b*c)^3/(d*x+c)^(3/2)-2/3*c*(a*d+b*
c)*(7*a^2*d^2-22*a*b*c*d+7*b^2*c^2)*x^2*(b*x+a)^(1/2)/b^2/d^2/(-a*d+b*c)^4/(d*x+c)^(1/2)-1/12*((a*d+b*c)*(105*
a^4*d^4-340*a^3*b*c*d^3+406*a^2*b^2*c^2*d^2-340*a*b^3*c^3*d+105*b^4*c^4)-2*b*d*(35*a^4*d^4-76*a^3*b*c*d^3+18*a
^2*b^2*c^2*d^2-76*a*b^3*c^3*d+35*b^4*c^4)*x)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4/d^4/(-a*d+b*c)^4

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {100, 155, 152, 65, 223, 212} \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{9/2} d^{9/2}}-\frac {2 c x^3 \sqrt {a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}-\frac {2 c x^2 \sqrt {a+b x} (a d+b c) \left (7 a^2 d^2-22 a b c d+7 b^2 c^2\right )}{3 b^2 d^2 \sqrt {c+d x} (b c-a d)^4}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )-2 b d x \left (35 a^4 d^4-76 a^3 b c d^3+18 a^2 b^2 c^2 d^2-76 a b^3 c^3 d+35 b^4 c^4\right )\right )}{12 b^4 d^4 (b c-a d)^4}+\frac {2 a x^4 (13 b c-7 a d)}{3 b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[In]

Int[x^6/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(2*a*x^5)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*(13*b*c - 7*a*d)*x^4)/(3*b^2*(b*c - a*d)^2*
Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b^2*c^2 + 14*a*b*c*d - 7*a^2*d^2)*x^3*Sqrt[a + b*x])/(3*b^2*d*(b*c - a*
d)^3*(c + d*x)^(3/2)) - (2*c*(b*c + a*d)*(7*b^2*c^2 - 22*a*b*c*d + 7*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*b^2*d^2*(b
*c - a*d)^4*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*((b*c + a*d)*(105*b^4*c^4 - 340*a*b^3*c^3*d + 406*a^
2*b^2*c^2*d^2 - 340*a^3*b*c*d^3 + 105*a^4*d^4) - 2*b*d*(35*b^4*c^4 - 76*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 76*
a^3*b*c*d^3 + 35*a^4*d^4)*x))/(12*b^4*d^4*(b*c - a*d)^4) + (5*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sq
rt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(9/2)*d^(9/2))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 a x^5}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {2 \int \frac {x^4 \left (5 a c+\frac {1}{2} (-3 b c+7 a d) x\right )}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{3 b (b c-a d)} \\ & = \frac {2 a x^5}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (13 b c-7 a d) x^4}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {4 \int \frac {x^3 \left (2 a c (13 b c-7 a d)+\frac {1}{4} \left (-3 b^2 c^2+62 a b c d-35 a^2 d^2\right ) x\right )}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{3 b^2 (b c-a d)^2} \\ & = \frac {2 a x^5}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (13 b c-7 a d) x^4}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+14 a b c d-7 a^2 d^2\right ) x^3 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {8 \int \frac {x^2 \left (\frac {9}{4} a c \left (b^2 c^2+14 a b c d-7 a^2 d^2\right )+\frac {3}{8} \left (7 b^3 c^3-9 a b^2 c^2 d+69 a^2 b c d^2-35 a^3 d^3\right ) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{9 b^2 d (b c-a d)^3} \\ & = \frac {2 a x^5}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (13 b c-7 a d) x^4}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+14 a b c d-7 a^2 d^2\right ) x^3 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {2 c (b c+a d) \left (7 b^2 c^2-22 a b c d+7 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {16 \int \frac {x \left (-\frac {3}{4} a c (b c+a d) \left (7 b^2 c^2-22 a b c d+7 a^2 d^2\right )-\frac {3}{16} \left (35 b^4 c^4-76 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-76 a^3 b c d^3+35 a^4 d^4\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{9 b^2 d^2 (b c-a d)^4} \\ & = \frac {2 a x^5}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (13 b c-7 a d) x^4}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+14 a b c d-7 a^2 d^2\right ) x^3 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {2 c (b c+a d) \left (7 b^2 c^2-22 a b c d+7 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right )-2 b d \left (35 b^4 c^4-76 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-76 a^3 b c d^3+35 a^4 d^4\right ) x\right )}{12 b^4 d^4 (b c-a d)^4}+\frac {\left (5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^4 d^4} \\ & = \frac {2 a x^5}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (13 b c-7 a d) x^4}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+14 a b c d-7 a^2 d^2\right ) x^3 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {2 c (b c+a d) \left (7 b^2 c^2-22 a b c d+7 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right )-2 b d \left (35 b^4 c^4-76 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-76 a^3 b c d^3+35 a^4 d^4\right ) x\right )}{12 b^4 d^4 (b c-a d)^4}+\frac {\left (5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b^5 d^4} \\ & = \frac {2 a x^5}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (13 b c-7 a d) x^4}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+14 a b c d-7 a^2 d^2\right ) x^3 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {2 c (b c+a d) \left (7 b^2 c^2-22 a b c d+7 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right )-2 b d \left (35 b^4 c^4-76 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-76 a^3 b c d^3+35 a^4 d^4\right ) x\right )}{12 b^4 d^4 (b c-a d)^4}+\frac {\left (5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b^5 d^4} \\ & = \frac {2 a x^5}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (13 b c-7 a d) x^4}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+14 a b c d-7 a^2 d^2\right ) x^3 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {2 c (b c+a d) \left (7 b^2 c^2-22 a b c d+7 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (105 b^4 c^4-340 a b^3 c^3 d+406 a^2 b^2 c^2 d^2-340 a^3 b c d^3+105 a^4 d^4\right )-2 b d \left (35 b^4 c^4-76 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-76 a^3 b c d^3+35 a^4 d^4\right ) x\right )}{12 b^4 d^4 (b c-a d)^4}+\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{9/2} d^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00 \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {-105 a^7 d^5 (c+d x)^2+5 a^6 b d^4 (47 c-28 d x) (c+d x)^2-3 a^5 b^2 d^3 (c+d x)^2 \left (22 c^2-106 c d x+7 d^2 x^2\right )-b^7 c^4 x^2 \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )+3 a^4 b^3 d^2 (c+d x)^2 \left (-22 c^3-32 c^2 d x+17 c d^2 x^2+2 d^3 x^3\right )-3 a b^6 c^3 x \left (70 c^4+15 c^3 d x-92 c^2 d^2 x^2-21 c d^3 x^3+8 d^4 x^4\right )+a^3 b^4 c d \left (235 c^5+186 c^4 d x-207 c^3 d^2 x^2-168 c^2 d^3 x^3-42 c d^4 x^4-24 d^5 x^5\right )+3 a^2 b^5 c^2 \left (-35 c^5+110 c^4 d x+183 c^3 d^2 x^2+4 c^2 d^3 x^3-14 c d^4 x^4+12 d^5 x^5\right )}{12 b^4 d^4 (b c-a d)^4 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{9/2} d^{9/2}} \]

[In]

Integrate[x^6/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(-105*a^7*d^5*(c + d*x)^2 + 5*a^6*b*d^4*(47*c - 28*d*x)*(c + d*x)^2 - 3*a^5*b^2*d^3*(c + d*x)^2*(22*c^2 - 106*
c*d*x + 7*d^2*x^2) - b^7*c^4*x^2*(105*c^3 + 140*c^2*d*x + 21*c*d^2*x^2 - 6*d^3*x^3) + 3*a^4*b^3*d^2*(c + d*x)^
2*(-22*c^3 - 32*c^2*d*x + 17*c*d^2*x^2 + 2*d^3*x^3) - 3*a*b^6*c^3*x*(70*c^4 + 15*c^3*d*x - 92*c^2*d^2*x^2 - 21
*c*d^3*x^3 + 8*d^4*x^4) + a^3*b^4*c*d*(235*c^5 + 186*c^4*d*x - 207*c^3*d^2*x^2 - 168*c^2*d^3*x^3 - 42*c*d^4*x^
4 - 24*d^5*x^5) + 3*a^2*b^5*c^2*(-35*c^5 + 110*c^4*d*x + 183*c^3*d^2*x^2 + 4*c^2*d^3*x^3 - 14*c*d^4*x^4 + 12*d
^5*x^5))/(12*b^4*d^4*(b*c - a*d)^4*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (5*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*
ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(4*b^(9/2)*d^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3424\) vs. \(2(401)=802\).

Time = 0.60 (sec) , antiderivative size = 3425, normalized size of antiderivative = 7.70

method	result	size
default	\(\text {Expression too large to display}\)	\(3425\)

[In]

int(x^6/(b*x+a)^(5/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/24*(-84*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^4*c^2*d^5*x^4-84*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b
^5*c^3*d^4*x^4-48*a^3*b^4*c*d^6*x^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+72*a^2*b^5*c^2*d^5*x^5*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)-48*a*b^6*c^3*d^4*x^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1098*a^5*b^2*c^2*d^5*x^2*((b*x+a
)*(d*x+c))^(1/2)*(b*d)^(1/2)+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a
^8*c^2*d^6+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^6*c^8+105*ln(
1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^8*d^8*x^2+105*ln(1/2*(2*b*d*x+2*((b
*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^8*c^8*x^2+1098*a^2*b^5*c^5*d^2*x^2*((b*x+a)*(d*x+c))^
(1/2)*(b*d)^(1/2)-414*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^4*c^4*d^3*x^2+660*(b*d)^(1/2)*((b*x+a)*(d*x+c)
)^(1/2)*a^6*b*c^2*d^5*x-90*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^6*b*c*d^6*x^2-414*(b*d)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)*a^4*b^3*c^3*d^4*x^2+126*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^6*c^4*d^3*x^4+372*(b*d)^(1/2)*((b*x+a
)*(d*x+c))^(1/2)*a^5*b^2*c^3*d^4*x-456*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^3*c^4*d^3*x+372*(b*d)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)*a^3*b^4*c^5*d^2*x+660*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^5*c^6*d*x-336*(b*d)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^4*c^3*d^4*x^3+24*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^5*c^4*d^3*x^3+24*(
b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^3*c^2*d^5*x^3-90*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^6*c^6*d*x^2+
552*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^6*c^5*d^2*x^3+552*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^5*b^2*c*d^
6*x^3+126*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^3*c*d^6*x^4-210*a^7*d^7*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)-210*b^7*c^7*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-210*a^7*c^2*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-
210*a^2*b^5*c^7*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+
a*d+b*c)/(b*d)^(1/2))*a^6*b^2*d^8*x^4+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)
^(1/2))*b^8*c^6*d^2*x^4+210*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^7*b*
d^8*x^3+210*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^8*c^7*d*x^3+210*ln(1
/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^8*c*d^7*x+210*ln(1/2*(2*b*d*x+2*((b*
x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^7*c^8*x-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^7*b*c^3*d^5+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*
d+b*c)/(b*d)^(1/2))*a^6*b^2*c^4*d^4+60*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1
/2))*a^5*b^3*c^5*d^3+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^4*c
^6*d^2-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^5*c^7*d+330*ln(1/
2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b^3*c^3*d^5*x^2+510*ln(1/2*(2*b*d*x
+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^4*c^4*d^4*x^2+330*ln(1/2*(2*b*d*x+2*((b*x+a
)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^5*c^5*d^3*x^2-840*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^6*c^6*d^2*x^2+150*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^7*c^7*d*x^2-330*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b
*c)/(b*d)^(1/2))*a^7*b*c^2*d^6*x-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2
))*a^6*b^2*c^3*d^5*x+390*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b^3*c
^4*d^4*x+390*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^4*c^5*d^3*x-270
*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^5*c^6*d^2*x-330*ln(1/2*(2*b
*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^6*c^7*d*x+470*(b*d)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)*a^6*b*c^3*d^4-132*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^5*b^2*c^4*d^3-132*(b*d)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)*a^4*b^3*c^5*d^2+470*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^4*c^6*d-42*(b*d)^(1/2)*((b*x+a)*(d*x+
c))^(1/2)*a^5*b^2*d^7*x^4-42*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^7*c^5*d^2*x^4-280*(b*d)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)*a^6*b*d^7*x^3-280*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^7*c^6*d*x^3+12*a^4*b^3*d^7*x^5*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)+12*b^7*c^4*d^3*x^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-420*a^7*c*d^6*x*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)-420*a*b^6*c^7*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b^3*c*d^7*x^4+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^4*c^2*d^6*x^4+60*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*
d+b*c)/(b*d)^(1/2))*a^3*b^5*c^3*d^5*x^4+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*
d)^(1/2))*a^2*b^6*c^4*d^4*x^4-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*
a*b^7*c^5*d^3*x^4-330*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^6*b^2*c*d^
7*x^3-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b^3*c^2*d^6*x^3+390*
ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^4*c^3*d^5*x^3+390*ln(1/2*(2*
b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^5*c^4*d^4*x^3-270*ln(1/2*(2*b*d*x+2*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^6*c^5*d^3*x^3-330*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^7*c^6*d^2*x^3+150*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*
(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^7*b*c*d^7*x^2-840*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*
d+b*c)/(b*d)^(1/2))*a^6*b^2*c^2*d^6*x^2)/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(a*d-b*c)^4/(b*x+a)^(3/2)/(d*x+c)
^(3/2)/b^4/d^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1470 vs. \(2 (402) = 804\).

Time = 2.45 (sec) , antiderivative size = 2954, normalized size of antiderivative = 6.64 \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(x^6/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(15*(7*a^2*b^6*c^8 - 18*a^3*b^5*c^7*d + 9*a^4*b^4*c^6*d^2 + 4*a^5*b^3*c^5*d^3 + 9*a^6*b^2*c^4*d^4 - 18*a
^7*b*c^3*d^5 + 7*a^8*c^2*d^6 + (7*b^8*c^6*d^2 - 18*a*b^7*c^5*d^3 + 9*a^2*b^6*c^4*d^4 + 4*a^3*b^5*c^3*d^5 + 9*a
^4*b^4*c^2*d^6 - 18*a^5*b^3*c*d^7 + 7*a^6*b^2*d^8)*x^4 + 2*(7*b^8*c^7*d - 11*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5*d^3
+ 13*a^3*b^5*c^4*d^4 + 13*a^4*b^4*c^3*d^5 - 9*a^5*b^3*c^2*d^6 - 11*a^6*b^2*c*d^7 + 7*a^7*b*d^8)*x^3 + (7*b^8*
c^8 + 10*a*b^7*c^7*d - 56*a^2*b^6*c^6*d^2 + 22*a^3*b^5*c^5*d^3 + 34*a^4*b^4*c^4*d^4 + 22*a^5*b^3*c^3*d^5 - 56*
a^6*b^2*c^2*d^6 + 10*a^7*b*c*d^7 + 7*a^8*d^8)*x^2 + 2*(7*a*b^7*c^8 - 11*a^2*b^6*c^7*d - 9*a^3*b^5*c^6*d^2 + 13
*a^4*b^4*c^5*d^3 + 13*a^5*b^3*c^4*d^4 - 9*a^6*b^2*c^3*d^5 - 11*a^7*b*c^2*d^6 + 7*a^8*c*d^7)*x)*sqrt(b*d)*log(8
*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) +
8*(b^2*c*d + a*b*d^2)*x) - 4*(105*a^2*b^6*c^7*d - 235*a^3*b^5*c^6*d^2 + 66*a^4*b^4*c^5*d^3 + 66*a^5*b^3*c^4*d
^4 - 235*a^6*b^2*c^3*d^5 + 105*a^7*b*c^2*d^6 - 6*(b^8*c^4*d^4 - 4*a*b^7*c^3*d^5 + 6*a^2*b^6*c^2*d^6 - 4*a^3*b^
5*c*d^7 + a^4*b^4*d^8)*x^5 + 21*(b^8*c^5*d^3 - 3*a*b^7*c^4*d^4 + 2*a^2*b^6*c^3*d^5 + 2*a^3*b^5*c^2*d^6 - 3*a^4
*b^4*c*d^7 + a^5*b^3*d^8)*x^4 + 4*(35*b^8*c^6*d^2 - 69*a*b^7*c^5*d^3 - 3*a^2*b^6*c^4*d^4 + 42*a^3*b^5*c^3*d^5
- 3*a^4*b^4*c^2*d^6 - 69*a^5*b^3*c*d^7 + 35*a^6*b^2*d^8)*x^3 + 3*(35*b^8*c^7*d + 15*a*b^7*c^6*d^2 - 183*a^2*b^
6*c^5*d^3 + 69*a^3*b^5*c^4*d^4 + 69*a^4*b^4*c^3*d^5 - 183*a^5*b^3*c^2*d^6 + 15*a^6*b^2*c*d^7 + 35*a^7*b*d^8)*x
^2 + 6*(35*a*b^7*c^7*d - 55*a^2*b^6*c^6*d^2 - 31*a^3*b^5*c^5*d^3 + 38*a^4*b^4*c^4*d^4 - 31*a^5*b^3*c^3*d^5 - 5
5*a^6*b^2*c^2*d^6 + 35*a^7*b*c*d^7)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^9*c^6*d^5 - 4*a^3*b^8*c^5*d^6 + 6*a
^4*b^7*c^4*d^7 - 4*a^5*b^6*c^3*d^8 + a^6*b^5*c^2*d^9 + (b^11*c^4*d^7 - 4*a*b^10*c^3*d^8 + 6*a^2*b^9*c^2*d^9 -
4*a^3*b^8*c*d^10 + a^4*b^7*d^11)*x^4 + 2*(b^11*c^5*d^6 - 3*a*b^10*c^4*d^7 + 2*a^2*b^9*c^3*d^8 + 2*a^3*b^8*c^2*
d^9 - 3*a^4*b^7*c*d^10 + a^5*b^6*d^11)*x^3 + (b^11*c^6*d^5 - 9*a^2*b^9*c^4*d^7 + 16*a^3*b^8*c^3*d^8 - 9*a^4*b^
7*c^2*d^9 + a^6*b^5*d^11)*x^2 + 2*(a*b^10*c^6*d^5 - 3*a^2*b^9*c^5*d^6 + 2*a^3*b^8*c^4*d^7 + 2*a^4*b^7*c^3*d^8
- 3*a^5*b^6*c^2*d^9 + a^6*b^5*c*d^10)*x), -1/24*(15*(7*a^2*b^6*c^8 - 18*a^3*b^5*c^7*d + 9*a^4*b^4*c^6*d^2 + 4*
a^5*b^3*c^5*d^3 + 9*a^6*b^2*c^4*d^4 - 18*a^7*b*c^3*d^5 + 7*a^8*c^2*d^6 + (7*b^8*c^6*d^2 - 18*a*b^7*c^5*d^3 + 9
*a^2*b^6*c^4*d^4 + 4*a^3*b^5*c^3*d^5 + 9*a^4*b^4*c^2*d^6 - 18*a^5*b^3*c*d^7 + 7*a^6*b^2*d^8)*x^4 + 2*(7*b^8*c^
7*d - 11*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5*d^3 + 13*a^3*b^5*c^4*d^4 + 13*a^4*b^4*c^3*d^5 - 9*a^5*b^3*c^2*d^6 - 11*
a^6*b^2*c*d^7 + 7*a^7*b*d^8)*x^3 + (7*b^8*c^8 + 10*a*b^7*c^7*d - 56*a^2*b^6*c^6*d^2 + 22*a^3*b^5*c^5*d^3 + 34*
a^4*b^4*c^4*d^4 + 22*a^5*b^3*c^3*d^5 - 56*a^6*b^2*c^2*d^6 + 10*a^7*b*c*d^7 + 7*a^8*d^8)*x^2 + 2*(7*a*b^7*c^8 -
11*a^2*b^6*c^7*d - 9*a^3*b^5*c^6*d^2 + 13*a^4*b^4*c^5*d^3 + 13*a^5*b^3*c^4*d^4 - 9*a^6*b^2*c^3*d^5 - 11*a^7*b
*c^2*d^6 + 7*a^8*c*d^7)*x)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/
(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(105*a^2*b^6*c^7*d - 235*a^3*b^5*c^6*d^2 + 66*a^4*b^4*c^5
*d^3 + 66*a^5*b^3*c^4*d^4 - 235*a^6*b^2*c^3*d^5 + 105*a^7*b*c^2*d^6 - 6*(b^8*c^4*d^4 - 4*a*b^7*c^3*d^5 + 6*a^2
*b^6*c^2*d^6 - 4*a^3*b^5*c*d^7 + a^4*b^4*d^8)*x^5 + 21*(b^8*c^5*d^3 - 3*a*b^7*c^4*d^4 + 2*a^2*b^6*c^3*d^5 + 2*
a^3*b^5*c^2*d^6 - 3*a^4*b^4*c*d^7 + a^5*b^3*d^8)*x^4 + 4*(35*b^8*c^6*d^2 - 69*a*b^7*c^5*d^3 - 3*a^2*b^6*c^4*d^
4 + 42*a^3*b^5*c^3*d^5 - 3*a^4*b^4*c^2*d^6 - 69*a^5*b^3*c*d^7 + 35*a^6*b^2*d^8)*x^3 + 3*(35*b^8*c^7*d + 15*a*b
^7*c^6*d^2 - 183*a^2*b^6*c^5*d^3 + 69*a^3*b^5*c^4*d^4 + 69*a^4*b^4*c^3*d^5 - 183*a^5*b^3*c^2*d^6 + 15*a^6*b^2*
c*d^7 + 35*a^7*b*d^8)*x^2 + 6*(35*a*b^7*c^7*d - 55*a^2*b^6*c^6*d^2 - 31*a^3*b^5*c^5*d^3 + 38*a^4*b^4*c^4*d^4 -
31*a^5*b^3*c^3*d^5 - 55*a^6*b^2*c^2*d^6 + 35*a^7*b*c*d^7)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^9*c^6*d^5 -
4*a^3*b^8*c^5*d^6 + 6*a^4*b^7*c^4*d^7 - 4*a^5*b^6*c^3*d^8 + a^6*b^5*c^2*d^9 + (b^11*c^4*d^7 - 4*a*b^10*c^3*d^8
+ 6*a^2*b^9*c^2*d^9 - 4*a^3*b^8*c*d^10 + a^4*b^7*d^11)*x^4 + 2*(b^11*c^5*d^6 - 3*a*b^10*c^4*d^7 + 2*a^2*b^9*c
^3*d^8 + 2*a^3*b^8*c^2*d^9 - 3*a^4*b^7*c*d^10 + a^5*b^6*d^11)*x^3 + (b^11*c^6*d^5 - 9*a^2*b^9*c^4*d^7 + 16*a^3
*b^8*c^3*d^8 - 9*a^4*b^7*c^2*d^9 + a^6*b^5*d^11)*x^2 + 2*(a*b^10*c^6*d^5 - 3*a^2*b^9*c^5*d^6 + 2*a^3*b^8*c^4*d
^7 + 2*a^4*b^7*c^3*d^8 - 3*a^5*b^6*c^2*d^9 + a^6*b^5*c*d^10)*x)]

Sympy [F]

\[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x^{6}}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**6/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Integral(x**6/((a + b*x)**(5/2)*(c + d*x)**(5/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^6/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1536 vs. \(2 (402) = 804\).

Time = 1.13 (sec) , antiderivative size = 1536, normalized size of antiderivative = 3.45 \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(x^6/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

1/12*((3*(b*x + a)*(2*(b^20*c^7*d^6 - 7*a*b^19*c^6*d^7 + 21*a^2*b^18*c^5*d^8 - 35*a^3*b^17*c^4*d^9 + 35*a^4*b^
16*c^3*d^10 - 21*a^5*b^15*c^2*d^11 + 7*a^6*b^14*c*d^12 - a^7*b^13*d^13)*(b*x + a)/(b^22*c^7*d^7*abs(b) - 7*a*b
^21*c^6*d^8*abs(b) + 21*a^2*b^20*c^5*d^9*abs(b) - 35*a^3*b^19*c^4*d^10*abs(b) + 35*a^4*b^18*c^3*d^11*abs(b) -
21*a^5*b^17*c^2*d^12*abs(b) + 7*a^6*b^16*c*d^13*abs(b) - a^7*b^15*d^14*abs(b)) - (7*b^21*c^8*d^5 - 32*a*b^20*c
^7*d^6 + 28*a^2*b^19*c^6*d^7 + 112*a^3*b^18*c^5*d^8 - 350*a^4*b^17*c^4*d^9 + 448*a^5*b^16*c^3*d^10 - 308*a^6*b
^15*c^2*d^11 + 112*a^7*b^14*c*d^12 - 17*a^8*b^13*d^13)/(b^22*c^7*d^7*abs(b) - 7*a*b^21*c^6*d^8*abs(b) + 21*a^2
*b^20*c^5*d^9*abs(b) - 35*a^3*b^19*c^4*d^10*abs(b) + 35*a^4*b^18*c^3*d^11*abs(b) - 21*a^5*b^17*c^2*d^12*abs(b)
+ 7*a^6*b^16*c*d^13*abs(b) - a^7*b^15*d^14*abs(b))) - 4*(35*b^22*c^9*d^4 - 195*a*b^21*c^8*d^5 + 420*a^2*b^20*
c^7*d^6 - 380*a^3*b^19*c^6*d^7 - 90*a^4*b^18*c^5*d^8 + 630*a^5*b^17*c^4*d^9 - 756*a^6*b^16*c^3*d^10 + 468*a^7*
b^15*c^2*d^11 - 153*a^8*b^14*c*d^12 + 21*a^9*b^13*d^13)/(b^22*c^7*d^7*abs(b) - 7*a*b^21*c^6*d^8*abs(b) + 21*a^
2*b^20*c^5*d^9*abs(b) - 35*a^3*b^19*c^4*d^10*abs(b) + 35*a^4*b^18*c^3*d^11*abs(b) - 21*a^5*b^17*c^2*d^12*abs(b
) + 7*a^6*b^16*c*d^13*abs(b) - a^7*b^15*d^14*abs(b)))*(b*x + a) - 3*(35*b^23*c^10*d^3 - 230*a*b^22*c^9*d^4 + 6
15*a^2*b^21*c^8*d^5 - 840*a^3*b^20*c^7*d^6 + 510*a^4*b^19*c^6*d^7 + 204*a^5*b^18*c^5*d^8 - 714*a^6*b^17*c^4*d^
9 + 696*a^7*b^16*c^3*d^10 - 369*a^8*b^15*c^2*d^11 + 106*a^9*b^14*c*d^12 - 13*a^10*b^13*d^13)/(b^22*c^7*d^7*abs
(b) - 7*a*b^21*c^6*d^8*abs(b) + 21*a^2*b^20*c^5*d^9*abs(b) - 35*a^3*b^19*c^4*d^10*abs(b) + 35*a^4*b^18*c^3*d^1
1*abs(b) - 21*a^5*b^17*c^2*d^12*abs(b) + 7*a^6*b^16*c*d^13*abs(b) - a^7*b^15*d^14*abs(b)))*sqrt(b*x + a)/(b^2*
c + (b*x + a)*b*d - a*b*d)^(3/2) + 8/3*(9*a^5*b^5*c^3*d - 23*a^6*b^4*c^2*d^2 + 19*a^7*b^3*c*d^3 - 5*a^8*b^2*d^
4 - 18*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^3*c^2*d + 27*(sqrt(b*d)*sqrt(b*
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^2*c*d^2 - 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^2*a^7*b*d^3 + 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^5*b*c*
d - 6*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^6*d^2)/((sqrt(b*d)*b^5*c^3*abs(b) -
3*sqrt(b*d)*a*b^4*c^2*d*abs(b) + 3*sqrt(b*d)*a^2*b^3*c*d^2*abs(b) - sqrt(b*d)*a^3*b^2*d^3*abs(b))*(b^2*c - a*b
*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^3) - 5/8*(7*b^2*c^2 + 10*a*b*c*d + 7*a
^2*d^2)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b^3*d^4*abs(b))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x^6}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]

[In]

int(x^6/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x)

[Out]

int(x^6/((a + b*x)^(5/2)*(c + d*x)^(5/2)), x)